The results of the probability theory help us to predict a lot of future events, to estimate our chances to win or to lose in many life situations. The beginning of this theory was connected with the investigations of the so-called Bernoulli random variables. These simplest variables take on only two values, but they describe many situations in our life. Up to now the different random schemes need to get and to use the new results for these two-valued random variables. The authors of this book want to give the readers useful information about the role of Bernoulli distributions in probability, in mathematical statistics and in their different applications.